Faltings's theorem

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field $\mathbb{Q}$ of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell,Template:Sfn and known as the Mordell conjecture until its 1983 proof by Gerd Faltings.Template:Sfnm The conjecture was later generalized by replacing $\mathbb{Q}$ by any number field.

Background

Let $C$ be a non-singular algebraic curve of genus $g$ over $\mathbb{Q}$. Then the set of rational points on $C$ may be determined as follows:

• When $g=0$, there are either no points or infinitely many. In such cases, $C$ may be handled as a conic section.
• When $g=1$, if there are any points, then $C$ is an elliptic curve and its rational points form a finitely generated abelian group. (This is Mordell's Theorem, later generalized to the Mordell–Weil theorem.) Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup.
• When $g>1$, according to Faltings's theorem, $C$ has only a finite number of rational points.

Proofs

Igor Shafarevich conjectured that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a fixed finite set of places.Template:Sfn Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.Template:Sfn

Gerd Faltings proved Shafarevich's finiteness conjecture using a known reduction to a case of the Tate conjecture, together with tools from algebraic geometry, including the theory of Néron models.Template:Sfn The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.Template:Efn

Later proofs

• Paul Vojta gave a proof based on Diophantine approximation.Template:Sfn Enrico Bombieri found a more elementary variant of Vojta's proof.Template:Sfn
• Brian Lawrence and Akshay Venkatesh gave a proof based on [[p-adic Hodge theory|Template:Mvar-adic Hodge theory]], borrowing also some of the easier ingredients of Faltings's original proof.Template:Sfn

Consequences

Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:

• The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points;
• The Isogeny theorem that abelian varieties with isomorphic Tate modules (as ${\mathbb{Q}}_{\ell }$-modules with Galois action) are isogenous.

A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed $n\ge 4$ there are at most finitely many primitive integer solutions (pairwise coprime solutions) to $a^n+b^n=c^n$, since for such $n$ the Fermat curve $x^n+y^n=1$ has genus greater than 1.

Generalizations

Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve $C$ with a finitely generated subgroup $\mathrm{\Gamma }$ of an abelian variety $A$. Generalizing by replacing $A$ by a semiabelian variety, $C$ by an arbitrary subvariety of $A$, and $\mathrm{\Gamma }$ by an arbitrary finite-rank subgroup of $A$ leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillanTemplate:Sfn following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.

Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if $X$ is a pseudo-canonical variety (i.e., a variety of general type) over a number field $k$, then $X(k)$ is not Zariski dense in $X$. Even more general conjectures have been put forth by Paul Vojta.

The Mordell conjecture for function fields was proved by Yuri Ivanovich ManinTemplate:Sfn and by Hans Grauert.Template:Sfn In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.Template:Sfn

References

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